US20150330918A1 - Method of determining surface orientation of single crystal wafer - Google Patents

Abstract

Provided is a method of determining a surface orientation of a single crystal wafer. The method of determining a surface orientation of a single crystal wafer using high resolution X-ray rocking curve measurement may determine a surface angle of the wafer and a direction of the surface angle using rocking curve measurement of a high resolution X-ray diffraction method and measuring a misalignment angle formed by a rotation axis of a measuring apparatus and a surface normal of the wafer and an orientation of the misalignment angle.

Description

TECHNICAL FIELD
• The present invention relates to a method of determining a surface orientation of a single crystal wafer, and more particularly, to a method of determining a surface orientation of a single crystal wafer using high resolution X-ray rocking curve measurement capable of determining a surface angle of the wafer and a direction of the surface angle using rocking curve measurement of a high resolution X-ray diffraction method and measuring a misalignment angle formed by a rotation axis of a measuring apparatus and a surface normal of the wafer and an orientation of the misalignment angle.
BACKGROUND ART
• A single crystal wafer of silicon, sapphire, gallium arsenide, or the like, for manufacturing a semiconductor device is manufactured so as to have a predetermined crystallographical directional property. In the case of the single crystal wafer, since general information on a surface orientation of the wafer is present and a single crystal is well processed, an axis orientation of the wafer may be determined using an X-ray.
• A general single crystal wafer has been produced according to standards such as an angle between a surface and a crystal plane of 0±0.5° and 4±0.5°, a horizontal component of a surface orientation of 0.2±0.05°, and a vertical component of a surface orientation of 0±0.1° with respect to a (100) wafer or a (111) wafer. As a usual single crystal wafer usually used as a material of a semiconductor device, a wafer of which a surface normal is tilted with respect to a silicon crystal plane normal by about 0 to 4° is used. Since this angle (a surface orientation, off-cut angle, surface miscut, or surface misorientation) and a direction (an off-cut or miscut direction) in which the surface normal of the wafer is tilted have an effect on a physical property of a manufactured semiconductor device, it is very important to accurately measure the surface orientation. In addition, since this angle and this direction are important factors in determining productivity of the device, they have been importantly controlled in a production line of the wafer for a semiconductor device.
• For these reasons, accuracy of an apparatus of measuring and inspecting the surface orientation becomes a decisive factor in determining quality of a product as well as productivity of the production line. Therefore, the apparatus of measuring the surface orientation of the wafer needs to be accurately calibrated before subsequent processing processes such as a polishing process, and the like.
• In order to accurately determine an accurate angle formed by the surface normal of the wafer and a vertical axis of the crystal plane and a direction of the angle, a measuring method using an X-ray diffractometer (hereinafter, referred to as an XRD) has been demanded.
• Meanwhile, a standard for measuring a crystallographical surface orientation of a single crystal wafer using the XRD has been defined in a standard procedure ASTM F26-87a (Standard Test Method for Determining the Orientation of a Semiconductive Single Crystal). In the ASTM F26-87a standard, which is a standard for measuring a crystallographical orientation of a semiconductive single crystal, a method using an XRD and an optical method have been described. In the method using an XRD, procedures such as an X-ray diffraction theory for measuring an orientation of a semiconductive single crystal, a measuring apparatus, a measuring method, an analyzing method, and the like, has been described.
• However, this standard has been described under the assumption that a surface normal of a wafer is the same as a rotation axis of a measuring apparatus. In a general case, since the surface normal of the wafer does not coincide with the rotation axis of the measuring apparatus, a measuring error corresponding to an angle at which they do not coincide with each other is caused. Therefore, in the case in which the surface normal of the single crystal wafer requiring a precise surface orientation is significantly different from the rotation axis of the measuring apparatus, large uncertainty is caused in measuring the surface orientation of the wafer.
DISCLOSURE Technical Problem
• An object of the present invention is to provide a method of determining a surface orientation of a single crystal wafer capable of precisely determining not only the surface orientation, that is, a surface angle, of the single crystal wafer but also a direction of the surface angle using rocking curve measurement of a high resolution X-ray diffraction method.
• Particularly, an object of the present invention is to provide a method capable of accurately determining a surface orientation of a wafer in consideration of misalignment between a rotation axis of an apparatus of measuring a surface orientation and a surface normal of the wafer without arranging the rotation axis and the surface normal using a surface orientation standard material even in the case in which the rotation axis and the surface normal do not coincide with each other and is to provide a method of determining a surface orientation of a single crystal wafer using high resolution X-ray rocking curve measurement capable of measuring an angle formed by a surface normal of the wafer and a rotation axis of a measuring apparatus and a direction of the angle.
Technical Solution
• In one general aspect, there is provided a method of measuring a surface orientation of a signal crystal wafer in order to determine the surface orientation formed by a crystal plane normal of a single crystal and a surface normal of the wafer, wherein the wafer is rotated by a predetermined rotation angle (θ) with respect to the surface normal thereof to measure a high-resolution X-ray rocking curve of a selected diffraction plane under an optimal Bragg diffraction condition, and a position (ω_φ) of a maximum peak of the high-resolution X-ray rocking curve is determined by the following Equation:
• 
  ω_φ=ω₀+Δωφ=θ_E+δ₀−δ_P(R)
• where ω₀ indicates an incident angle of an X-ray, θ_E indicates a Bragg angle, δ₀ indicates a misalignment angle formed by a rotation axis and the surface normal, and δ_P(R) indicates an angle between a rotation axis of a measuring apparatus and a surface normal on a diffraction plane rotated by
• A tilt angle (δS(R)) of the surface normal of the wafer may be determined by the following Equation:
• 
  δ_S(R)≅δ₀·cos(φ−φ_δ)−δ_S(R)
• where δ_δindicates a phase of the surface normal, and δ_S(R) indicates a geometrical small angle component.
• At the rotation angle φ=0 the position (ω_φ) of the maximum peak of the high-resolution X-ray rocking curve may be determined by the following Equation:
• 
  ω_φ≅θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)−δ_P(R)
• where φ_δindicates a phase of the surface normal, and κ_S(R) indicates a geometrical small angle component.
• An angle (δ_P(R)) between the rotation axis and the crystal plane normal having a function on the diffraction plane may be determined by the following Equation:
• 
  δ_P(R)≅δ₁·cos(φ−φ_δ)+δ₀·cos(φ−φ_δ)−κ_P(R)
• where δ₁·cos(φ−φ_δ) indicates an angle component of the crystal plane normal changed along a circumference of the surface normal, and δ₀·cos(φ−φ_δ) indicates an angle component of the surface normal changed along a circumference of the rotation axis.
• When considering misalignment of the rotation axis rotating the wafer, the position (ω_φ) of the maximum peak of the high-resolution X-ray rocking curve may be determined by the following Equation:
• 
  ω_φ≅δ₁·cos(φ−φ_δ)−δ₀·cos(φ−φ_δ)+κ_P(R)+θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)
• where δ₁ indicates an angle (surface angle) of the crystal plane normal with respect to the surface normal, φ_δindicates a direction in which the surface angle appears, δ₀ indicates a misalignment angle formed by the rotation axis and the surface normal, φ_δindicates a direction of a misalignment axis, κ_P(R) indicates a small angle component, θ_B indicates a Bragg angle, and the remainings indicate constants.
• The surface angle (δ₁) of the wafer and the direction (φ_δ) in which the surface angle appears may be determined by the following Equation:
• ${\omega }_{\phi }-{\omega }_{\phi }^{\prime }\cong 2{\delta }_1·\sin \frac{{\mathrm{\Delta \phi }}_p}{2}·\sin \left(\phi -{\phi }_p-\frac{{\mathrm{\Delta \phi }}_p}{2}\right)$
• where Δφ_δindicates a phase change value applied at the time of designing a wafer holder, and ω_φ−ω′_φ indicates an angle difference between peaks of the high-resolution X-ray rocking curve each measured depending on Δφ_δ, and
• a variation (δ_P(S)) of the surface orientation of the wafer depending on a function of the orientation angle (θ) may be determined by the following Equation:
• 
  δ_P(S)≅δ₁·cos(φ−φ_δ).
• The high-resolution X-ray rocking curve may be measured two times at φ=φ₁ and φ=φ₂, and Δφ_δmay be determined by the following Equation: Δφ_δ=φ₂−φ₁.
• A tilt variation (δ_S(R)) of the surface normal from the rotation axis depending on the function of the orientation angle (δ) may be determined by the following Equation:
• 
  δ_S(R)≅δ₀·cos(φ−φ_δ)
• where δ₀ indicates the misalignment angle formed by the rotation axis and the surface normal, and φ_δindicates the direction of the misalignment axis.
• An angle component (δ₁·cos φ_δ) of the surface orientation of the wafer along a direction of 0 to 180° may be determined by the following Equation:
• 
  δ₁·cos φ_δ=½(ω′₀−ω₀),
• an angle component (δ₁·sin φ_δ) of the surface orientation of the wafer along a direction of 90 to 270° may be determined by the following Equation:
• 
  δ₁·sin φ_δ=½(ω′_δ0−ω_δ0), and
• the surface orientation of the single crystal wafer may be measured only by measuring the high-resolution X-ray rocking curve two times at an interval of 90° at each of the two sample orientations of Δφ_δ=180°.
Advantageous Effects
• With the method of determining a surface orientation of a single crystal wafer using high resolution X-ray rocking curve measurement according to an exemplary embodiment of the present invention having the configuration as described above, the direction of the surface angle as well as the surface angle of the wafer is accurately determined, thereby making it possible to contribute to improvement of productivity of the wafer and improve quality of a product. In addition, even in the case in which the rotation axis of the measuring apparatus for measuring the surface orientation of the wafer and the surface normal of the wafer do not coincide with each other, the surface orientation of the wafer may be accurately determined without arranging the rotation axis and the surface normal using a surface orientation standard material. In addition, the misalignment angle formed by the rotation axis of the measuring apparatus and the surface normal and the direction of the misalignment angle may be determined.
DESCRIPTION OF DRAWINGS
• FIG. 1 is a diagram showing a reciprocal lattice space for a single crystal wafer having a crystal plane of a surface orientation angle δ₁ when a surface normal n_S is parallel to a rotation axis n_R of a measuring apparatus;
• FIG. 2 is a diagram showing a reciprocal lattice space for a single crystal wafer having a crystal plane of a surface orientation angle δ₁ when the surface normal n_S is tilted with respect to the rotation axis n_R of the measuring apparatus by a tilt angle δ₀;
• FIG. 3 is a diagram showing a variation of κ_P(S) on a diffraction plane in the case (See FIG. 1) in which the surface normal is parallel to the rotation axis when a Bragg diffraction condition for a reciprocal lattice point R is satisfied;
• FIG. 4 is a diagram showing a result of a κ_P(S) value when δ_P(S) changes from δ₁ to −δ₁ for a sample with δ₁=1.0°;
• FIG. 5 is a diagram showing a result of a κ_P(S) value calculated as a function of an azimuth angle for δ₁=1.0° when it is assumed that a phase angle φ_δ=0;
• FIG. 6 is a diagram showing a sample wafer having a miscut of δ₁=0.201° and φ_δ=9.59° from a reference edge; and
• FIG. 7 is a diagram showing the least-square-fit of ω_φ−ω′_φ by a sine function.
BEST MODE
• Hereinafter, an exemplary embodiment of the present invention of the present invention will be described in detail with reference to the accompanying drawings.
• Theoretical Model
• FIG. 1 shows a reciprocal lattice space for a single crystal wafer having a surface orientation angle δ₁ when a surface normal n_S is parallel to a rotation axis n_R of a measuring apparatus. A reciprocal lattice point for a reflection plane selected in order to measure a rocking curve is shown at a point P along a direction of a crystal plane normal n_P.
• A diffraction plane containing an incident X-ray beam and a reflected X-ray beam is put in a vertical direction, which is a direction of a paper surface, and passes through an original point O, as shown in FIG. 1. k₀ indicates a wave vector of an incident X-ray, and k_B a indicates a wave vector of a reflected X-ray. In FIG. 1, it has been assumed that the crystal plane normal is accurately put on the diffraction plane at φ=1. Therefore, it has been assumed that an optimal Bragg condition is satisfied without any x rotation.
• When an incidence angle of an X-ray is ω₀ at an azimuth angle φ=0, the Bragg's law is satisfied for the reciprocal lattice point P. When ignoring refractive index corrections, an incidence angle ω₀ in which a peak of the rocking curve appears is represented by the following Equation.
• $\begin{array}{cc}{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="US20150330918A1-20151119-D00002.png,US20150330918A1-20151119-D00004.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0=\frac{\pi }{2}-\alpha -{\delta }_1={\theta }_B-{\delta }_1& \left(\mathrm{Equation}1\right)\end{array}$
• Where
• $\alpha =\frac{\pi }{2}-{\theta }_B,$
• and θ_B indicates a Bragg angle. When the wafer is rotated with respect to the surface normal n_R by φ=φ₀, the point P moves to a point Q along a cone having a semi-apex angle of δ₁. The point is, then, rotated along an axis x by x=x_δin order to satisfy an accurate Bragg reflection condition and moves to a point R on the diffraction plane. The incident angle ω_φ of the lattice point R satisfying the Bragg's law, that is, a peak position of the rocking curve is represented by the following Equation.
• 
  ω_φ=ω₀+Δω_φ−δ_P(S)=θ_B−δ_P(S)  (Equation 2)
• From right triangles ΔOA′Q, ΔOA′R′, and ΔA′R′Q shown in FIG. 1, the following Equation may be derived.
• 
  tan+(δ_P(S)+κ_P(S))=tan δ₁·cos φ_δ  (Equation 3)
• In addition, when δ₁>1, Equation 3 may be represented by the following Equation.
• 
  δ_P(S)≅δ₁·cos φ_δ−κ_P(S)  (Equation 4)
• Here, Equation 4 may be generalized as the following Equation in consideration of the phase φ_δof the crystal plane normal at which the azimuth angle φ is rotated by the rotation axis through φ−φ_δ.
• 
  δ_P(S)≅δ₁·cos(φ−φ_δ)−κ_P(S)  (Equation 5)
• δ_P(S) is a function of an angle and indicates an angle between the crystal plane normal and the surface normal on the diffraction plane, and κ_P(S) varies depending on φ. In addition, from Equations 2 and 5, a variation in the incident angle depending on a variation in the azimuth angle φ is represented by the following Equation.
• 
  ω_φ≅δ₁·cos(φ−φ_δ)+θ_B+κ_P(S)  (Equation 6)
• In the case in which the surface normal of the sample is not parallel to the rotation axis of the measuring apparatus, a situation becomes more complicated. FIG. 2 shows a reciprocal lattice space for a single crystal wafer having a crystal plane of a surface miscut δ₁ when the surface normal n_S is not parallel to the rotation axis n_R of the measuring apparatus.
• δ₀ indicates a misalignment angle of the surface normal n_S with respect to the rotation axis n_R. In FIG. 2, it has been assumed that the crystal plane normal and the surface normal are accurately put on the plane of diffraction at φ=0. Therefore, it has been assumed that an optimal Bragg condition is satisfied without any x rotation.
• Referring to FIG. 2, when the reciprocal lattice point P is rotated by φ=φ_δ, the reciprocal lattice point P moves to a point δ. The point then moves to a point r on the diffraction plane through x=x_q rotation. At the same time, the surface normal n_S also rotates by φ=φ_δand x=x_δ. In this case, since δ₁=δ₀, the surface normal is not present on the diffraction plane. n_S′ in FIG. 2 indicates projection of the surface normal n_S′ on the diffraction plane after the surface normal rotates by φ=φ_δand x=x_δ. When the surface normal rotates by φ=φ_δand x=x_δ, respectively, the position of the sample surface varies from S₀ to S_φ, as shown in FIG. 2.
• In the case in which the Bragg's law is satisfied for the reciprocal lattice point P at the azimuth angle φ=₀, the incidence angle ω₀ is given as ω₀=θ_B−δ₁. As shown in FIG. 2, the incident angle ω_φ, that is, the peak position of the rocking curve is defined by the following Equation by a Bragg condition for the lattice point r by and x rotation.
• 
  ω_φ=ω_φ+Δω_φ=θ_B+δ₀−δ_P(R)  (Equation 7)
• Where δω_φ=δ₁+δ₀−δ_P(R). By analogy with Equation 5, the tilt angle between the surface normal and the rotation axis on the diffraction plane may be represented as a function of φ by the following Equation.
• 
  δ_S(R)≅δ₀·cos(φ−φ_δ)−κ_S(R)  (Equation 7-1)
• Where φ_δis a phase for the movement as in Equation 5. δ₀ in Equation 7 indicates the tilt angle of the surface normal from the rotation axis at φ=0 under the condition in shown in FIG. 2.
• When using Equation 7-1, the tilt of the surface normal at φ=0 is generalized as δ_S(R)(φ=0)=δ₀·cos(−φ_δ)κ_S(R)(φ=0). Therefore, Equation 7 may be represented by the following Equation.
• 
  ω_φ≅θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)−δ_P(R)  (Equation 8)
• From right triangles δOα′α, Δα′r′, and Δα′r′α shown in FIG. 2 and the condition as in Equation 3, the following Equation may be derived.
• 
  tan(δ_P(R)+κ_P(R)=tan(δ₁+δ₀)·cos φ_δ  (Equation 9)
• For δ₁+δ₀<1, Equation 9 may be represented by the following Equation.
• 
  δ_P(R)≅δ₁·cos(φ_δ+δ₀·cos φ_δ−κ_P(R)  (Equation 10)
• Equation 10 is also generalized by Equation 5, and each cosine function as in Equation 10 may be represented by the following Equation when considering any phases φ_δand φ_δdefined by Equations 5 and 7-1.
• 
  δ_P(R)≅δ₁·cos(φ−φ_δ)+δ₀·cos(φ−φ_δ)−_κP(R)  (Equation 11)
• δ_P(R) indicates an angle between the crystal plane normal and the rotation axis as a function of δon the diffraction plane. In addition, κ_P(R) varies as the function of φ. δ₁·cos(φ−φ_P) indicates a movement component of the crystal plane normal along of a circumference of the surface normal as in Equation 5, δ₀·cos(φ−φ_δ) and indicates a movement component of the surface normal along a circumference of the rotation axis as in Equation 7-1.
• When inserting Equation 11 into Equation 8, the following Equation may be derived.
• 
  ω_φ≅δ₁·cos(φ−φ_δ)−δ₀·cos(φ−φ_δ)+κ_P(R)+θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)  (Equation 12)
• Equation 13 may describe a variation of the peak position of the rocking curve for the selected reflection plane as a function of the azimuth angle δeven though the surface normal is not parallel to the rotation axis. Therefore, when δ₀ is equal to zero, Equation 13 becomes Equation 6.
• In Equation 6, when the phase φ_δof the cosine function changes by Δφ_δ, the variation of the incidence angle is represented by the following Equation.
• 
  ω′_φ≅−δ₁·cos(φ−φ_δ−Δφ_δ)+θ_B+κ′_P(S)  (Equation 13)
• From Equations 6 and 13, the following Equation may be derived.
• $\begin{array}{cc}{\omega }_{\phi }-{\omega }_{\phi }^{\prime }=2{\delta }_1·\sin \left(\frac{{\mathrm{\Delta \phi }}_p}{2}\right)·\sin \left(\phi -{\phi }_p-\frac{{\mathrm{\Delta \phi }}_p}{2}\right)+{\kappa }_{P\left(S\right)}-{\kappa }_{P\left(S\right)}^{\prime }& \left(\mathrm{Equation}14\right)\end{array}$
• Similar to Equation 12, even though the phase φ_δchanges by φ_δ+Δφ_δ, the phase φ_δrelated to the movement of δ_a is maintained in a fixed state. Therefore, Equation 12 may be represented by the following Equation.
• 
  ω′_φ≅−δ₁·cos(φ−φ_δ−Δφ_δ)−δ₀·cos(φ−φ_δ)+κ′_P(R)+θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)  (Equation 15)
• From Equations 12 and 15, the following Equation may be derived.
• $\begin{array}{cc}{\omega }_{\phi }-{\omega }_{\phi }^{\prime }=2{\delta }_1·\sin \left(\frac{{\mathrm{\Delta \phi }}_p}{2}\right)·\sin \left(\phi -{\phi }_p-\frac{{\mathrm{\Delta \phi }}_p}{2}\right)+{\kappa }_{P\left(R\right)}-{\kappa }_{P\left(R\right)}^{\prime }& \left(\mathrm{Equation}16\right)\end{array}$
• FIG. 3 shows a variation of κ_P(S) on a diffraction plane in the case (See FIG. 1) in which the surface normal is parallel to the rotation axis when a Bragg diffraction condition a reciprocal lattice point 3 is satisfied. As shown in FIG. 3, the following Equation is satisfied from ΔOA′R′.
• $\begin{array}{cc}\tan \left({\delta }_{P\left(S\right)}+{\kappa }_{P\left(S\right)}\right)=\frac{\sin {\delta }_{P\left(S\right)}}{\cos {\delta }_1}& \left(\mathrm{Equation}17\right)\end{array}$
• Therefore, κ_P(S) may be given as the following Equation.
• $\begin{array}{cc}{\kappa }_{P\left(S\right)}={\tan }^{-1}\left(\frac{\sin {\delta }_{P\left(S\right)}}{\cos {\delta }_1}\right)-{\delta }_{P\left(S\right)}& \left(\mathrm{Equation}18\right)\end{array}$
• When inserting Equation 5 into Equation 18, the following Equation may be derived.
• $\begin{array}{cc}{\kappa }_{P\left(S\right)}\cong {\tan }^{-1}\left(\frac{\sin \left({\delta }_1·\cos \left(\phi -{\phi }_p\right)-{\kappa }_{P\left(S\right)}\right)}{\cos {\delta }_1}\right)-{\delta }_1·\cos \left(\phi -{\phi }_p\right)+{\kappa }_{P\left(S\right)}& \left(\mathrm{Equation}19\right)\end{array}$
• Equation 19 shows the variation of κ_P(S) as a function of φ when the surface normal is parallel to the rotation axis.
Experimental Example
• A surface orientation was measured for a 6 inch (00.1) sapphire wafer used as a substrate for a light emitting diode (LED) and having a nominal surface azimuth angle of 0.2° using the theoretical models described above. A high resolution X-Ray diffractometer (XRD) including a 4-bounce Ge (022) monochromator and a 4-circle goniometer were utilized as the measuring apparatus. In addition, the surface of the wafer was closely attached to a reference surface of a wafer holder. The wafer holder includes a narrow and long slit with different two azimuth angles φ₁ and φ₂ having an angle difference of 120° therebetween. The two slits were configured to be parallel to reference edges of the wafer at φ₁ and φ₂, respectively.
• Measurements of rocking curves were carried out as follows. At any one azimuth angle of the sample waver at φ=φ₁, rocking curves of an optimum Bragg conditions for a sapphire (00.6) crystal plane were measured six times at each of the different azimuths at an interval of 60° (for example, φ=0, 60, 120, 180, 240

  

  360). In addition, the peak position of each rocking curve was recorded. After the measurements at φ=φ₁, the wafer was removed from the holder and the sample was again fixed to the holder so that φ=φ₂. Then, the rocking curves were measured six times as described above. Before rocking curve measurements, an azimuth angle of the wafer mounted on the holder were accurately determined. The narrow and long slit of the holder parallel to the reference edge of the wafer was aligned to be parallel to a direction of the X-ray through a φ scan. In this case, the peak position was determined as φ=φ₁. After rotating the wafer to φ=φ₂, another slit in the holder was aligned to be parallel to the direction of the X-ray through a φ scan. In this case, the peak position was determined as φ=φ₂. A difference of Δφ_δ=φ₂−φ₁ becomes a phase change for the surface orientation measurement.
• Result
• Assuming that the phase angle φ_δ=0, the angle δ_P(S) between the crystal plane normal and the surface normal on the diffraction plane shown in FIG. 1 varies from δ₁ to −δ₁ when the azimuth angle rotates from φ to δ. κ_P(S) is calculated according to Equation 19 when δ_P(S) changes from δ₁ to δ₁ for a sample with δ₁=1.0°. The result is shown in FIG. 4. Maximum and minimum values of κ_P(S) are ±5.9×10⁻⁵(°) and the values of κ_P(S) are negligibly small as compared to the surface angle δ₁=1.0°.
• Table 1 shows maximum and minimum values of κ_P(S) calculated according to Equation 18 for samples with δ₁=0.2, 1.0, 1.5, 2.0, 2.5 and 3.0. It could be appreciated that the maximum and minimum values are very small as compared to the surface angles.

• TABLE 1
  δ1	κP(S) (°)	κP(S) − κ′P(S) (°)
  (°)	max. (+)/min.(−)	max (+)/min. (−)
  0.2	±4.7 × 10−7	±4.7 × 10−7
  1.0	±5.9 × 10−5	±5.9 × 10−5
  1.5	±2.0 × 10−4	±2.0 × 10−4
  2.0	±4.7 × 10−4	±4.7 × 10−4
  2.5	±9.2 × 10−4	±9.2 × 10−4
  3.0	±1.6 × 10−3	±1.6 × 10−3

• Therefore, since κ_P(S) is extremely smaller than δ₁, κ_P(S) at the right side of Equation 19 can be neglected. Therefore, Equation 19 may be represented by the following Equation.
• $\begin{array}{cc}{\kappa }_{P\left(S\right)}\cong {\tan }^{-1}\left(\frac{\sin \left({\delta }_1·\cos \left(\phi -{\phi }_p\right)\right)}{\cos {\delta }_1}\right)-{\delta }_1·\cos \left(\phi -{\phi }_p\right)& \left(\mathrm{Equation}20\right)\end{array}$
• When assuming a phase φ_δ=0, κ_P(S) is calculated as a function of the azimuth angle for δ₁=1.0° according to Equation 20. The result is shown in FIG. 5. The variation is symmetric around φ=180°, and the value of κ_P(S) is 0 when φ=0, 90, 180 and 270 as shown in FIG. 1. When a phase of a cosine function in Equation 20 shifts by Δφ_δ, a difference κ_P(S)−κ′_P(S) between two values in Equation 14 for the wafer having δ₁=1.0° has the smallest maximum value of 5.9×10⁻⁵(°) when Δφ_δ=125°. The value in this case is equal to the maximum value of κ_P(S). The variations of κ′_P(S) and κ_P(S)−κ′_P(S), which are a function of the azimuth angle φ for the phase change of Δφ_δ=125°, are shown in FIG. 5 together with the variation of κ_P(S). The maximum and minimum values in κ_P(S)−κ′_P(S) values when Δφ_δ=125° for different δ₁ values are also shown in Table 1 together with those in the respective κ_P(S) values. In this experiment, a phase shift of Δφ_δ=125° was employed at the time of designing the wafer holder and the difference κ_P(S)−κ′_P(S) for the phase change when Δφ_δ=125° was neglected during analysis.
• As described above, the rocking curves under the optimum Bragg conditions for the sapphire (00.6) crystal plane were measured six times at a sample azimuth angle φ=φ₁ and were additionally measured six times at a sample azimuth angle φ=φ₂. Table 2 shows the peak positions of the rocking curves at each azimuth In addition, the measured phase change Δφ_δ=φ₂−φ₁ was 120.19°.

• TABLE 2
  δP(S) = OCA, δS(R) = δ0 · cos(φ − φs) (Δφp = 120.19)

  φ (°)	ωφ	ω′φ	ωφ − ω′φ	δP(S)	ωφ + δ P(S)

  0	20.6318	20.9591	−0.3273	0.1979	20.8297
  60	20.6797	20.7389	−0.0592	0.1279	20.8076
  90	20.7664
  120	20.8680	20.5985	0.2695	−0.0700	20.7980
  180	21.0074	20.6816	0.3258	−0.1979	20.8095
  240	20.9587	20.9013	0.0574	−0.1279	20.8308
  270	20.8744
  300	20.7753	21.0408	−0.2655	0.0700	20.8453

• Since the rotation axis of a goniometer is not usually parallel to the surface normal of the sample, Equation 16 was used toe analyze the surface orientation. From ω_φ and ω′_φ in Table 2, ω_φ−ω′_φ is calculated at each φ. This is also shown in Table 2. ω_φ−ω′_φ at each may be fitted to a sine function using the least squares method according to Equation 16. The term κ_P(R)−κ′_P(R) may be neglected. The reason is that the value is negligibly small in the case in which δ₁≈0.2°, as shown in Table 1. Therefore, ω_φ−ω′_φ is determined by the following Equation.
• $\begin{array}{cc}{\omega }_{\phi }-{\omega }_{\phi }^{\prime }\cong 2{\delta }_1·\sin \frac{{\mathrm{\Delta \phi }}_p}{2}·\sin \left(\phi -{\phi }_p-\frac{{\mathrm{\Delta \phi }}_p}{2}\right)=0.348·\sin \left(\phi -69.69\right)+0.0001& \left(\mathrm{Equation}21\right)\end{array}$
• According to Equation 21, δ₁=0.201° and φ_δ=9.59° may be obtained using Δφ_δ=120.19°. Therefore, the sample wafer has a surface miscut of δ₁=0.201° at φ_δ=9.59° from the reference edge of the sample. The result is schematically shown in FIG. 6. From Equation 5, the variation δ_P(S) of the surface orientation of the wafer as the function of φ may be defined by the following Equation.
• 
  δ_P(S)≅0.201·cos(φ−9.59)  (Equation 22)
• The least squares fit in Equation 21 for ω_φ−ω′_φ by a sine function is shown in FIG. 7. The value of x² for the fitting is 3.7×10⁻⁶ corresponding to an extremely small value, which shows that the confidence level of the fit is very high.
• Using Equation 5, Equation 12 may be rewritten as the following Equation.
• 
  ω_φ+δ_P(S)≅−δ₀·cos(φ−φ_δ)+θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)+κ_P(R)−κ_P(S)  (Equation 23)
• δ_P(S) and ω_φ+δ_P(S) at each φ are calculated together with the measurement values of the peak positions ω_φ and ω′_φ as shown in Table 2. According to Equation 23, the value of ω_φ+δ_P(S) is fitted to a cosine function as a function of φ using the least squares method. The result is represented by the following Equation.
• 
  ω_φ+δ_P(S)≅−0.023·cos(φ−297.85)+20.820  (Equation 24)
• Therefore, the variation δ_S(R) of the tilt of the surface normal from the rotation axis as a function of φ is determined as represented by the following Equation.
• 
  δ_S(R)≅0.023·cos(φ−117.85)  (Equation 25)
• Where when neglecting the term −κ_S(R)(φ=0)+κ_P(R)−κ_P(S) in Equation 23, Equation 25 shows that the maximum misalignment δ₀=0.023° at φ_δ=117.85°.
• Horizontal and vertical components of the surface orientation of the sample wafer were measured according to the ASTM standard (ASTM F26-87a, Standard Test Method for Determining the Orientation of a Semiconductive Single Crystal) and were compared to the results of the present experiment. When neglecting κ_P(R), at δ₁ and δ₀, Equation 12 may be defined by the following Equations, respectively.
• 
  ω₀≅−δ_P(S)(φ=0)−δ_S(R)(φ=0)+θ_B+δ₀·cos(−φ_S)−κ_S(R)(φ=0)  (Equation 26)
• 
  ω₁₈₀≅−δ_P(S)(φ=180)−δ_S(R)(φ=180)θ_B+δ₀·cos(−φ_S)−κ_S(R)(φ=0)  (Equation 27)
• From Equations 26 and 27, the following Equation may be derived.
• $\begin{array}{cc}\frac{{\omega }_{180}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="US20150330918A1-20151119-D00002.png,US20150330918A1-20151119-D00004.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0}{2}=-\frac{{\delta }_{P\left(S\right)\left(\phi =180\right)}-{\delta }_{P\left(S\right)\left(\phi =0\right)}}{2}-\frac{{\delta }_{S\left(R\right)\left(\phi =180\right)}-{\delta }_{S\left(R\right)\left(\phi =0\right)}}{2}& \left(\mathrm{Equation}28\right)\end{array}$
• Where the value
• $\frac{\left({\omega }_{180}-{\omega }_0\right)}{2}$
• by the ASTM may be obtained from at ω_φ at φ=0° and 180° in Table 2, and Equation 28 may be defined by the following Equation.
• $\begin{array}{cc}\frac{\mathrm{\Delta \omega }}{2}+\frac{{\mathrm{\Delta \delta }}_{S\left(R\right)}}{2}=-\frac{{\mathrm{\Delta \delta }}_{P\left(S\right)}}{2}& \left(\mathrm{Equation}29\right)\end{array}$
• Table 3 shows comparison results between the ASTM method and the present experiment and the relation between the two values that may be obtained according to Equation 29. The vertical component along a direction of 90°˜270° by the ASTM method is obtained from ω_φ at φ=90° and 270° in Table 2. Since the ASTM method does not incorporate the misalignment δ_S(R) of the surface normal from the rotation axis, Δω/2 and −Δδ_P(S)/2 are not consistent with each other. However, when δ_S(R) is incorporated in Δω/2, Δω/2+Δδ_S(R)/2 and −Δδ_P(S)/2 are almost equal to each other within measurement errors.

• 	TABLE 3
  	Surface	−ΔδP(S)/		ΔδS(R)/	(ΔδS(R) +
  	orientation(°)	2	Δω/2	2	Δω)/2

  	Horizontal(0-180)	0.198	0.188	0.011	0.199
  	Vertical(90-270)	0.033	0.054	−0.020	0.034

• In the present experiment, the rocking curves were measured six times per 60° at two different sample azimuth angles, that is, were measured twelve times, in order to increase precision of the fitting. However, it is sufficient in obtaining the surface orientation and the misalignment of the surface normal to measure the rocking curves four times per 90° at two different sample azimuth angles, that is, to measure the rocking curves eight times. In addition, in the present experiment, when the number of measurements of the rocking curves is increased, the precision of the analysis may be further increased.
• As described in Equation 25, the surface normal of the sample used in the present experiment has a maximum misalignment angle, that is, tilt angle δ₀=0.023° at φ_δ=117.85° from the rotation axis defined in the goniometer. In order to adjust the misalignment angle between the surface normal and the rotation axis of the goniometer, the misalignment angle was carefully adjusted by −0.023° at φ=117.85° to make the surface normal δ₀=0° of the sample. Then, the measurement was again performed.
• Table 4 shows results of the measurement values, and the variation of the surface orientation of the sample as a function of φ is represented by the following Equation.
• 
  δ_P(S)≅0.201·cos(φ−10.06)  (Equation 30)

• TABLE 4
  δP(S) = OCA, δS(R) = δ0 · cos(φ − φs) (Δφp = 120.06)

  φ (°)	ωφ	ω′φ	ωφ − ω′φ	δP(S)	ωφ + δ P(S)

  0	20.6327	20.9610	−0.3283	0.1980	20.8307
  60	20.6961	20.7576	−0.0615	0.1294	20.8255
  120	20.8917	20.6241	0.2676	−0.0686	20.8231
  180	21.0247	20.6977	0.3270	−0.1980	20.8267
  240	20.9612	20.9008	0.0604	−0.1294	20.8318
  300	20.7679	20.0332	−0.2653	0.0686	20.8365

• Equation 30 is almost the same as the result in Equation 22. The variation, that is, the misalignment, of the tilt of the surface normal from the rotation axis, is represented by the following Equation.
• 
  δ_S(R)≅0.006·cos(φ−114.03)  (Equation 31)
• The adjusted maximum tilt of the surface normal is δ₀=0.006°, and the tilt is very small as compared to an original value δ₀=0.023° before the adjustment.
• The value θ_B+δ₀·cos(−φ_S)−κ_S(R)(φ=0) in Equation 23 may be obtained from the measurement results in Table 2 and 4 are equal to 20.820° and 20.829° when neglecting κ_S(R)(φ=0). Bragg angles for the used diffraction plane are 20.831° and 20.832°. The two Bragg angles are almost the same as each other within experimental errors and are comparable to the theoretical value of 20.838° for a crystal plane (00.6) of sapphire.
• L. D. Doucette (L. D. Doucette et al, Review of Scientific Instruments 76, 036106, 2005) measured surface orientations for several single crystal wafers having miscuts of 5° or less by measuring rocking curves four times per 90° as a function of an azimuth angle at each of the two different sample azimuths (Δφ_δ=180°), that is, by measuring the rocking curves eight times. They considered the tilt angle δ_S(R) of the surface normal by Equation 7-1 in the present experiment with respect to the rotation axis.
• When neglecting κ_P(R) and κ′_P(R) and using Equations. 12 and 15, a horizontal component of the surface orientation along a direction of 0 to 180° is obtained. The horizontal component has the following relationship with Equation 4 of L. D. Doucette.
• $\begin{array}{cc}{\delta }_1·\cos {\phi }_p=\frac{1}{4}\left\{\left({\omega }_{180}-{\omega }_0\right)-\left({\mathrm{<figure-callout\; id="180"\; label="\omega "\; filenames="US20150330918A1-20151119-D00005.png,US20150330918A1-20151119-D00007.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{180}^{\prime }-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="US20150330918A1-20151119-D00002.png,US20150330918A1-20151119-D00004.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0^{\prime }\right)\right\}=-\frac{1}{4}\left\{\left({\omega }_1^{\prime }-{\omega }_2^{\prime }\right)+\left({\omega }_{2t}^{\prime }-{\omega }_{1t}^{\prime }\right)\right\}& \left(\mathrm{Equation}32\right)\end{array}$
• Where the sign relations are different, but the two results are equivalent to each other.
• In the case of a wafer of a very small surface miscut, in order to calculate the horizontal component δ₁·cos φ_δof the surface orientation along the direction of 0 to 180°, it is sufficient to measure the rocking curves only time times at two sample azimuths of Δφ_δ=180°. Therefore, when neglecting κ_P(R) and κ′_P(R) and using Equations. 12 and 15, an angular component along 0 to 180° is represented by the following Equation.
• 
  δ₁·cos φ_δ=½(ω′₀−ω₀)  (Equation 33)
• Similarly, an angular component along 90 to 270° is represented by the following Equation.
• 
  δ₁·sin φ_δ=½(ω′₉₀−ω₉₀)  (Equation 34)
• Therefore, the rocking curves are measured only two times at an interval of 90° at each of two sample azimuths of Δφ_δ=180°, that is, are measured only four times, thereby making it possible to calculate the surface orientation of the single crystal wafer.
CONCLUSION
• According to the exemplary embodiment of the present invention, theoretical models to completely describe the variation of the peak positions of the rocking curve as the function of the azimuth angle in both of the cases that the surface normal of the wafer is parallel and is not parallel to the rotation axis of the goniometer have been proposed. Based on these models, an accurate measurement method for the surface orientation of a single crystal wafer having a small surface miscut less than 3° has been proposed through rocking curve measurements using a high-resolution XRD. According to the exemplary embodiment of the present invention, it is possible to calculate the misalignment angle of the surface normal of the same with respect to the rotation axis of the goniometer as well as the surface orientation of the wafer. The surface orientation has been measured for a 6 inch sapphire wafer used for an LED substrate in the present invention. The surface orientation was measured to be δ₁=0.201° at φ=9.59° from the reference edge of the wafer in a clockwise direction. In addition, the misalignment of the surface normal from the rotation axis was measured to be δ₀=0.023° at φ=117.85, and was re-adjusted to 0.006°. During the analysis, geometrical angle components κ_P(S) and κ_P(S)−κ′_P(S) were calculated as the function of the azimuth angle and were negligibly small for the wafer having the surface miscut less than 3°. Surface orientations determined by the ASTM method were compared to the result values obtained by the present invention. The two results were consistent with each other when considering the tilt angle, that is, the misalignment, formed by the rotation axis and the surface normal. Finally, a method capable of simply and accurately calculating the surface orientation of the wafer by measuring the rocking curves two times at an interval of 90° at each of the two sample azimuths having a difference of 180°, that is, by measuring the rocking curves four times has been proposed.
• The present invention should not be construed to being limited to the above-mentioned exemplary embodiment. The present invention may be applied to various fields and may be variously modified by those skilled in the art without departing from the scope of the present invention claimed in the claims. Therefore, it is obvious to those skilled in the art that these alterations and modifications fall in the scope of the present invention.

Claims (9)

1. A method of measuring a surface orientation of a signal crystal wafer in order to determine the surface orientation formed by a crystal plane normal of a single crystal and a surface normal of the wafer, wherein the wafer is rotated by a predetermined rotation angle (φ) with respect to the surface normal thereof to measure a high-resolution X-ray rocking curve of a selected diffraction plane under an optimal Bragg diffraction condition, and a position (ω_φ) of a maximum peak of the high-resolution X-ray rocking curve is determined by the following Equation:

ω_φ=ω₀+Δωφ=θ_B+δ₀−δ_P(R)

where ω₀ indicates an incident angle of an X-ray, θ_B indicates a Bragg angle, δ₀ indicates a misalignment angle formed by a rotation axis and the surface normal, and δ_P(R) indicates an angle between a rotation axis of a measuring apparatus and a surface normal on a diffraction plane rotated by φ.

2. The method of claim 1, wherein a tilt angle (δ_S(R)) of the surface normal of the wafer is determined by the following Equation:

δ_S(R)≅δ₀·cos(φ−φ_δ)−κ_S(R)

where φ_δindicates a phase of the surface normal, and κ_S(R) indicates a geometrical small angle component.

3. The method of claim 1, wherein at the rotation angle φ=0, the position (ω_φ) of the maximum peak of the high-resolution X-ray rocking curve is determined by the following Equation:

ω_φ≅θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)−δ_P(R)

where φ_δindicates a phase of the surface normal, and κ_S(R) indicates a geometrical small angle component.

4. The method of claim 1, wherein an angle (δ_P(R)) between the rotation axis and the crystal plane normal having a function of φ on the diffraction plane is determined by the following Equation:

δ_P(R)≅δ₁·cos(φ−φ_δ)+δ₀·cos(φ−φ_δ)−κ_P(R)

where δ₁·cos(φ−φ_δ) indicates an angle component of the crystal plane normal changed along a circumference of the surface normal, and δ₀·cos(φ−φ_δ) indicates an angle component of the surface normal changed along a circumference of the rotation axis.

5. The method of claim 1, wherein when considering misalignment of the rotation axis rotating the wafer, the position (ω_φ) of which the maximum peak of the high-resolution X-ray rocking curve is determined by the following Equation:

ω_φ≅δ₁·cos(φ−φ_δ)−δ₀·cos(φ−φ_δ)+κ_P(R)+θ_B+δ₀·cos(−φ_δ)−κ_S(R)(φ=0)

where δ₁ indicates an angle (surface angle) of the crystal plane normal with respect to the surface normal, φ_δindicates a direction in which the surface angle appears, δ₀ indicates a misalignment angle formed by the rotation axis and the surface normal, φ_δindicates a direction of a misalignment axis, κ_P(R) indicates a small angle component, θ_B indicates a Bragg angle, and the remainings indicate constants.

6. The method of claim 5, wherein the surface angle (δ₁) of the wafer and the direction (φ_δ) in which the surface angle appears are determined by the following Equation:

${\omega }_{\phi }-{\omega }_{\phi }^{\prime }\cong 2{\delta }_1·\sin \frac{{\mathrm{\Delta \phi }}_p}{2}·\sin \left(\phi -{\phi }_p-\frac{{\mathrm{\Delta \phi }}_p}{2}\right)$

where Δφ_δindicates a phase change value applied at the time of designing a wafer holder, and ω_φ−ω′_φ indicates an angle difference between peaks of the high-resolution X-ray rocking curve each measured depending on Δφ_δ, and

a variation (δ_P(S)) of the surface orientation of the wafer depending on a function of the orientation angle (φ) is determined by the following Equation:

δ_P(S)≅δ₁·cos(φ−φ_δ).

7. The method of claim 6, wherein the high-resolution X-ray rocking curve is measured two times at φ=φ₁ and φ=φ₂, and Δφ_δis determined by the following Equation:

Δφ_δ=φ₂−φ₁.

8. The method of claim 6, wherein a tilt variation (δ_S(R)) of the surface normal from the rotation axis depending on the function of the orientation angle (φ) is determined by the following Equation:

δ_S(R)≅δ₀·cos(φ−φ_δ)

where δ₀ indicates the misalignment angle formed by the rotation axis and the surface normal, and φ_δindicates the direction of the misalignment axis.

9. The method of claim 6, wherein an angle component (δ₁·cos φ_δ) of the surface orientation of the wafer along a direction of 0 to 180° is determined by the following Equation:

δ₁·cos φ_δ=½(ω′₀−ω₀),

an angle component (δ₁·sin φ_δ) of the surface orientation of the wafer along a direction of 90 to 270° is determined by the following Equation:

δ₁·sin φ_δ=½(ω′₉₀−ω₉₀), and

the surface orientation of the single crystal wafer is measured only by measuring the high-resolution X-ray rocking curve two times at an interval of 90° at each of the two sample orientations of Δφ_δ=180°.

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